Complex Number
Complex Number Calculator
Add, subtract, multiply, divide and raise complex numbers to powers, with results in rectangular and polar form.
i² = −1
Also called complex calculator · imaginary number calculator
Math · 10 calculators
Complex Number Calculators
Arithmetic with complex numbers, conjugates, modulus, argument and conversion between polar and rectangular forms.
Complex Number
Complex Addition Calculator
Add two complex numbers (a + bi) + (c + di) by adding real and imaginary parts separately.
(a + bi) + (c + di) = (a + c) + (b + d)i
Also called add complex numbers · sum of complex numbers
Complex Number
Complex Subtraction Calculator
Subtract complex numbers (a + bi) − (c + di) and see the result in rectangular and polar form.
(a + bi) − (c + di) = (a − c) + (b − d)i
Also called subtract complex numbers · difference of complex numbers
Complex Number
Complex Multiplication Calculator
Multiply two complex numbers using FOIL and i² = −1, with the product in rectangular and polar form.
(a + bi)(c + di) = (ac − bd) + (ad + bc)i
Also called multiply complex numbers · product of complex numbers
Complex Number
Complex Division Calculator
Divide complex numbers by multiplying by the conjugate of the denominator; result in a + bi and polar form.
(a + bi)/(c + di) = [(ac + bd) + (bc − ad…
Also called divide complex numbers · complex quotient
Complex Number
Complex Conjugate Calculator
Find the complex conjugate of a + bi, and the product z·z̄ (always a real number equal to |z|²).
conj(a + bi) = a − bi
Also called conjugate of complex number · z bar
Complex Number
Polar Form Calculator
Convert a complex number from rectangular form a + bi to polar form r∠θ and exponential form re^(iθ).
r = √(a² + b²)
Also called rectangular to polar · complex to polar
Complex Number
Rectangular Form Calculator
Convert a complex number from polar form r∠θ to rectangular form a + bi.
a = r·cos θ
Also called polar to rectangular · polar to cartesian complex
Complex Number
Complex Magnitude Calculator
Find the modulus (absolute value) |z| of a complex number — its distance from the origin.
|a + bi| = √(a² + b²)
Also called modulus of complex number · absolute value of complex number
Complex Number
Complex Argument Calculator
Find the argument (angle) of a complex number in degrees and radians using atan2, with the correct quadrant.
arg(z) = atan2(b, a), in (−180°, 180°]
Also called arg of complex number · phase angle
Key complex number formulas
| Complex Number | i² = −1 · z = a + bi = r(cos θ + i sin θ) |
|---|---|
| Complex Addition | (a + bi) + (c + di) = (a + c) + (b + d)i |
| Complex Subtraction | (a + bi) − (c + di) = (a − c) + (b − d)i |
| Complex Multiplication | (a + bi)(c + di) = (ac − bd) + (ad + bc)i |
| Complex Division | (a + bi)/(c + di) = [(ac + bd) + (bc − ad)i] / (c² + d²) |
| Complex Conjugate | conj(a + bi) = a − bi |
| Polar Form | r = √(a² + b²) · θ = atan2(b, a) |
| Rectangular Form | a = r·cos θ · b = r·sin θ |
| Complex Magnitude | |a + bi| = √(a² + b²) |
| Complex Argument | arg(z) = atan2(b, a), in (−180°, 180°] |